Skip to main content
SpringerLink
Log in

Valdivia compact Abelian groups

Grupos abelianos compactos de Valdivia

  • Published:
RACSAM - Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas Aims and scope Submit manuscript

Abstract

LetR denote the smallest class of compact spaces containing all metric compacta and closed under limits of continuous inverse sequences of retractions. ClassR is striclty larger than the class of Valdivia compact spaces. We show that every compact connected Abelian group which is a topological retract of a space from classR is necessarily isomorphic to a product of metric compacta. This completes the result of Uspenskij and the author, where a compact Abelian group outside classR has been described.

Resumen

SeaR la clase mínima de espacios compactos que contiene a todos los espacios compactos métricos y que es cerrada para la operación de tomar límites de sucesiones continuas inversas de retracciones. La claseR es estrictamente mayor que la clase de los espacios compactos de Valdivia. Demostramos que todo grupo abeliano compacto conexo que es un retracto topológico de un espacio de la claseR es necesariamente isomorfo al producto de espacios métricos compactos. Este resultado completa el de Uspenskij y el autor, en el que se describe un grupo compacto abeliano que no pertenece a la claseR.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Argyros, S., Mercourakis, S. andNegrepontis, S., (1988). Functional-analytic properties of Corsoncompact spaces,Studia Math.,89, 3, 197–229.

    MATH  MathSciNet  Google Scholar 

  2. Burke, M., Kubiś, W. andTodorčević, S., (2006). Kadec norms on spaces of continuous functions,Serdica Math. J.,32, 2–3, 227–258. (Preprint available at http://arxiv.org/abs/math.FA/0312013)

    MATH  MathSciNet  Google Scholar 

  3. Chigogidze, A., (2008). Valdivia compact groups are products,Topology Appl.,155, 6, 605–609.

    Article  MATH  MathSciNet  Google Scholar 

  4. Deville, R. andGodefroy, G., (1993). Some applications of projective resolutions of identity,Proc. London Math. Soc.,(3) 67, 1, 183–199.

    Article  MathSciNet  Google Scholar 

  5. Deville, R., Godefroy, G. andZizler, V., (1993)Smoothness and Renormings in Banach Spaces. Pitman Monographs and Surveys in Pure and Applied Mathematics,64. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York.

    MATH  Google Scholar 

  6. Dow, A., (1988). An introduction to applications of elementary submodels to topology,Topology Proc.,13, 1, 17–72.

    MATH  MathSciNet  Google Scholar 

  7. Fabián, M., (1997)Gâteaux differentiability of convex functions and topology. Weak Asplund spaces, Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York.

    MATH  Google Scholar 

  8. Godement, R., (1964)Topologie algébrique et théorie des faisceaux, Hermann, Paris.

    Google Scholar 

  9. Hewitt, E. andRoss, K., (1963)Abstract Harmonic Analysis, Vol. I, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg,

    MATH  Google Scholar 

  10. Hofmann, K. H. andMorris, S. A., (1998)The structure of compact groups. A primer for the student — a handbook for the expert, de Gruyter Studies in Mathematics,25. Walter de Gruyter & Co., Berlin.

    MATH  Google Scholar 

  11. Kalenda, O., (1999). Embedding of the ordinal segment [0, ω1]into continuous images of Valdivia compacta,Comment. Math. Univ. Carolin.,40, 4, 777–783.

    MATH  MathSciNet  Google Scholar 

  12. Kalenda, O., (1999). Continuous images and other topological properties of Valdivia compacta, Fund. Math.162, 2, 181–192.

    MATH  MathSciNet  Google Scholar 

  13. Kalenda, O., (2000). Valdivia compact spaces in topology and Banach space theory,Extracta Math.,15, 1, 1–85.

    MATH  MathSciNet  Google Scholar 

  14. Kalenda, O., (2000). A characterization of Valdivia compact spaces,Collect. Math.,51, 1, 59–81.

    MATH  MathSciNet  Google Scholar 

  15. Kalenda, O., (2003). On the class of continuous images of Valdivia compacta,Extracta Math.,18, 1, 65–80.

    MATH  MathSciNet  Google Scholar 

  16. Kubiś, W., (2006). Compact spaces generated by retractions,Topology Applt.,153, 18, 3383–3396. (Preprint available at http://arxiv.org/abs/math/0511567)

    Article  MATH  Google Scholar 

  17. Kubiś, W. andMichalewski, H., (2006). Small Valdivia compact spaces,Topology Appl.,153, 14, 2560–2573. (Preprint available at http://arxiv.org/abs/math/0507062)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kubiś, W. andUspenskij, V., (2005). A compact group which is not Valdivia compact,Proc. Amer. Math. Soc.,133, 8, 2483–2487. (Preprint available at http://arxiv.org/abs/math/0311015)

    Article  MATH  MathSciNet  Google Scholar 

  19. Spanier, E., (1966)Algebraic Topology, McGraw-Hill, New York et al.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instytut Matematyki, Akademia Świętokrzyska, ul. Świętokrzyska 15, 25-406, Kielce, Poland

    Wieslaw Kubiś

Authors
  1. Wieslaw Kubiś
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wieslaw Kubiś.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kubiś, W. Valdivia compact Abelian groups. Rev. R. Acad. Cien. Serie A. Mat. 102, 193–197 (2008). https://doi.org/10.1007/BF03191818

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03191818

Keywords

Mathematics Subject Classifications

Search

Navigation